Fast Pairwise Conversion of Supernova Neutrinos: A Dispersion Relation Approach

Collective pair conversion νₑνₑ↔νₓνₓ by forward scattering, where x=μ or τ, may be generic for supernova neutrino transport. Depending on the local angular intensity of the electron lepton number carried by neutrinos, the conversion rate can be ``fast,'' i.e., of the order of √2GF(n_νₑ-n_νₑ)Δmₐₜₘ²/2E. We present a novel approach to understand these phenomena: a dispersion relation for the frequency and wave number (Ω,K) of disturbances in the mean field of νₑνₓ flavor coherence. Runaway solutions occur in ``dispersion gaps,'' i.e., in ``forbidden'' intervals of Ω and/or K where propagating plane waves do not exist. We stress that the actual solutions also depend on the initial and/or boundary conditions, which need to be further investigated.